Abstract: The Army’s myriad social networks connect not only humans and soldiers, but also machines, computers, and robots. And most of the social/biological/physical/informational connections in these networks are in the form of cooperation – entities working together to achieve a common goal. So how does this complex web of relationships, collaborations, and communities of diverse entities work? What are the metrics for a network and its social and informational value? What is cooperation and what makes a cooperative system effective? What forms of communication within a network can enhance cooperation? How do we measure trust and selflessness? Can our network structures, processes, tools, connections, communications, and languages enhance cooperation and optimize our networks? This paper reveals the role of mathematics as a tool to study these questions about networks and cooperative systems and builds a theory to formulate the fundamental principles, relationships, and metrics of these phenomena.